What is the equation for level curves in an elliptical shape?

In summary, the level curves for the function z=(5x^2+y^2)^.5-2x are elliptical in shape. This can be proven by setting z as a constant and using the equation of a standard ellipse, Ax^2+By^2=R^2, to algebraically manipulate the given function. The resulting equation, z^2+4xz=x^2+y^2, can be solved using the general equation for a conic, Ax^2+Bxy+Cy^2+Dx+Ey+F=0. Two suggestions for solving the equation are to think of z as a number and to use the equation of a general conic.
  • #1
wil3
179
1

Homework Statement



Describe the shape of each level curve for the following function:

z= (5x^2+y^2)^.5-2x


Homework Equations



I would like to prove that the curves are elliptical by setting z as a constnat and algebraically putting the equation in standard for for an ellipse Ax^2+By^2=R^2


The Attempt at a Solution



After squaring both sides, I get to:

z^2+4xz=x^2+y^2

I do not know how to isolate the z on one side from there. Any suggestions? I feel like this is a really obvious algebra trick that I am forgetting.

Thank you very much for any help.
 
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  • #2
Two suggestions:

- think of z as a number (that is arbitrary). No reason to isolate it

- Use the equation of the general conic Ax2+Bxy+Cy2+Dx+Ey+F=0
 

Related to What is the equation for level curves in an elliptical shape?

1. What is an ellipse?

An ellipse is a type of geometric shape that is defined as a closed curve in a plane, where the sum of the distances from any point on the curve to two fixed points (known as the foci) is constant.

2. How do you simplify an ellipse equation?

To simplify an ellipse equation, you can use the standard form which is (x-h)^2/a^2 + (y-k)^2/b^2 = 1, where (h,k) is the center of the ellipse and a and b are the lengths of the semi-major and semi-minor axes respectively. This form allows for easily identifying the center, vertices, and co-vertices of the ellipse.

3. What is the importance of simplifying an ellipse equation?

Simplifying an ellipse equation allows for a better understanding of the shape and its properties. It also makes it easier to graph and manipulate the equation for different calculations and applications.

4. Can you simplify an ellipse equation with non-integer coefficients?

Yes, an ellipse equation can be simplified with non-integer coefficients. It is important to note that the values of a and b in the standard form should be positive, and if necessary, you can factor out any common factors to simplify the equation further.

5. Are there any other forms of an ellipse equation?

Yes, there are two other forms of an ellipse equation - the center-radius form (x-h)^2/a^2 + (y-k)^2/b^2 = r^2 and the general form Ax^2 + By^2 + Cx + Dy + E = 0. These forms may be useful in different scenarios, but the standard form is the most commonly used and easiest to work with.

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